Some people referred me to the area = 1/2 height x base rule, but this is something my class doesn't know and don't need to know yet. (It seems to me further on than the thing we're trying to prove.)

John Golden suggested on Twitter that we see how you can get same-area triangles (if you start with the same "base"):> We also had a look at this:This convinced a lot of the students that Maryam's claim was true.

But Miguel was still not convinced. He said:
"Technology isn't perfect. Is that really a proof?"
Yes, he really did say that! Christmas came early to 4G.

We had a really good discussion about this, and quite a few students tried to persuade Miguel. There were a number of really good arguments to show how the two parts of the triangle in fact have equal areas. Hannah, for instance, suggested further splitting the pink parallelogram and the green parallelogram with their other diagonals. A lot of people thought this helped to see it.

After this, I asked the students to write about what had happened with this claim, to write what they thought about it all. Mostly great results:

Tuesday, 29 March 2016

I thought I'd document today's fraction talk lesson in a bit more detail, as we did a longer, 45-minute lesson on this rather than our usual ten minutes.

We started off working with a partner, looking at Beatriz's design:

One pair was measuring.

Some pairs made a little triangle

I circulated asking for explanations, and chipping in here and there. For instance:

Then we came together and shared what we'd decided:

We were divided in our opinion about whether the top right triangles were the same fraction or not (this was what I was hoping for), and so I moved on to Maria's square. This time, straightforward for most of the square, but what about those two on the left? Opinion was even more divided.

Then Maryam made her claim: "As long as it's the midpoint, it doesn't matter how you split the triangle; it will always be half." Yay! That will go on the claims board, and it's even more what I was hoping might happen!

To help people consider her claim, I drew this on the whiteboard:

Some students said it wasn't clear because of my drawing. OK then, I said, how about I draw it on Geogebra?

I used the midpoint tool. Maryam was still persuaded the two half-triangles were the same.

This time most people thought they were different. I said perhaps we'd need to come back to this. K. said that we needed to call Albert Einstein back from the grave to tell us. I was doubtful about both the possibility and necessity of that.

I was planning for them tomorrow to have a look at this image, made on an online geoboard, looking generally at all the fractions, and later focusing on the yellow triangle:

But, now Maryam has framed her claim differently to how I had, perhaps this diagram would be a better one to anatomise:

I think maybe we can see that whichever way we cut up the yellow triangle, we cut it in half.

Saturday, 5 March 2016

As my teaching evolves to start with students' thinking and talking, I'm usually beginning my hour long maths lessons with routines that encourage these things. Here are five favourites:

Quick Dot Images

See this by Steve Wyborney. I show an image quickly, then show it a second time; the idea is that almost all students should be able to have seen how many dots there are. But the interesting thing is not how many, but how. Students really like sharing their ways. And the message is: this is something you can do your way; you didn't need to be told how to see this. And - your way is worth sharing.

Students like to do the annotating themselves.
Often I do it: to speed things up, and to model revoicing.

Which One Doesn’t Belong

The idea is that you can justify the choice of any of the four items as the odd one out.

Estimate

Students write three numbers on a blank number line on their whiteboard, one too big, one too small and as close as you can (Graham Fletcher has some; these can be extended into whole lesson activities which are great modelling activities - and then of course there’s our regular, estimation180).

See, think, wonder

Last year it was notice and wonder, but I like the differentiation of notice into see and think, and feel it's worth distinguishing the two. (Both are brilliant though!) I'm reading Making Thinking Visible, and this is one of their routines.

The point is just to present an image and see what people see, preferably recording it all.

I've also adapted this to be what equations do you see, often presenting an image of Cuisenaire rods:

Counting Circles

I've blogged about this before. The idea is to count from different starting points, with different jumps. We do it round in a circle. Then we stop, and think about what the number three or four jumps on would be. Students share how they know.

NEW!Fraction Talks

I haven't used these yet, but this is a great idea, and Nat Banting is developing a really useful site with lots of images: fractiontalks.com - watch this space!